Negativity compensation in the nonnegative inverse eigenvalue problem
A set \(\Lambda \) of complex numbers is \textit{realizable} if it is the spectrum of an entrywise nonnegative matrix. \textit{R. L. Soto} [Linear Algebra Appl. 369, 169--184 (2003; Zbl 1031.15018)] proved that it is possible a real set \(\Lambda =\Lambda _{1}\cup \dots \cup \Lambda _{k}\) be realizable, without all the sets \(\Lambda _{1},\dots ,\Lambda _{k}\) be realizable. In the paper under review, the authors extend this result, allowing the sets to be complex. The main tool is a result due to \textit{A. Brauer} [Duke Math. J. 19, 75--91 (1952; Zbl 0046.01202); Theorem 3.1] then they associate to any nonrealizable set \(\Gamma \) a certain negativity \({\mathcal N}(\Gamma )\) and to any realizable set \(\Gamma \) a certain negativity \({\mathcal M}(\Gamma )\). A result of \textit{Kh. R. Suleĭmanova} [Dokl. Akad. Nauk SSSR, n. Ser. 66, 343--345 (1949; Zbl 0035.20903)] is also generalized, considering the complex set \(\{z\in \mathbb C :Re\, z\leq 0\), \(| Re\, z| \geq | Im\, z| \}\) that includes the negative real semi-axis. Some examples are given in the real and complex cases.
scientific article; zbMATH DE number 2126343
scientific article